Statistical Self-Similarity of One-Dimensional Growth Processes

Michael Praehofer; Herbert Spohn

doi:10.1016/s0378-4371(99)00517-8

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arXiv · cond-mat/9910273

Statistical Self-Similarity of One-Dimensional Growth Processes

Michael Praehofer·Herbert Spohn

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Abstract

For one-dimensional growth processes we consider the distribution of the height above a given point of the substrate and study its scale invariance in the limit of large times. We argue that for self-similar growth from a single seed the universal distribution is the Tracy-Widom distribution from the theory of random matrices and that for growth from a flat substrate it is some other, only numerically determined distribution. In particular, for the polynuclear growth model in the droplet geometry the height maps onto the longest increasing subsequence of a random permutation, from which the height distribution is identified as the Tracy-Widom distribution.

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Michael Praehofer, Herbert Spohn. 1999-10-24. Statistical Self-Similarity of One-Dimensional Growth Processes. https://doi.org/10.1016/s0378-4371(99)00517-8

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